Symmetry, Structure and Order Parameters

As in the nematics, the orientational order of the molecules is described by the order parameter (P2(cos )), where P2 is the second-order Legendre polynomial, 6 is the angle between the long axis of the molecule and the director, and ( ) denotes the average value. A sim- 

In the case of nematics, Priestley has described how the orientational distribution function could be expanded in a series of even-order Legendre polynomials  

How can this formal treatment of the distribution function (and resulting order parameters) be generalized to include the smectic-A structure We find the clue in Kirkwood s treatment of the melting of crystalline solids. In a crystal the density distribution function (the translational molecular distribution function) is periodic in three dimensions and can be expanded in a three-dimensional Fourier series. Kirkwood does this and then identifies the order parameters of the crystalline phase as the coefficients in the Fourier series. For simplicity let us consider a one-dimensionally periodic structure (such as the smectic-A but with the orientational order suppressed for the moment). The distribution function, which describes the tendency of the centers of mass of molecules to lie in layers perpendicular to the z-direction, can be expanded in a Fourier series  

The integral on the right hand side is immediately recognized as the definition of the average value, so that 

For the special case of m = 0, ao = 1/d. Combining these results we obtain 

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